Divisors and Sandpiles

This page provides access to a rough draft of a forthcoming textbook by Scott Corry and David Perkinson on the combinatorial theory of chip-firing related to the discrete Laplacian of a finite graph.

Contents
Part 1. Divisors
Chapter 1. The dollar game
  • An initial game
  • Formal definitions
  • The Picard and Jacobian groups
  • Notes
  • Problems
Chapter 2. The Laplacian
  • The discrete Laplacian
  • Configurations and the reduced Laplacian
  • Complete linear systems and convex polytopes
  • Structure of the Picard group
  • Notes
  • Problems
Chapter 3. Algorithms for winning
  • Greed
  • q-reduced divisors
  • Superstable configurations
  • Dhar's algorithm and efficient implementation
  • The Abel-Jacobi map
  • Notes
  • Problems
Chapter 4. Acyclic orientations
  • Orientations and maximal unwinnables
  • Dhar revisited
  • Notes
  • Problems
Chapter 5. Riemann-Roch
  • The rank function
  • Riemann-Roch for graphs
  • The analogy with Riemann surfaces
  • Alive divisors and stability
  • Notes
  • Problems
Part 2. Sandpiles
Chapter 6. The sandpile group
  • A first example
  • Directed graphs
  • Sandpile graphs
  • The reduced Laplacian
  • Recurrent sandpiles
  • Notes
  • Problems
Chapter 7. Burning and duality
  • Burning sandpiles
  • Existence and uniqueness
  • Superstables and recurrents
  • Forbidden configurations
  • Notes
  • Problems
Chapter 8. The threshold density theorem
  • Markov Chains
  • The fixed-energy sandpile
  • Main result
  • Notes
  • Problems
Part 3. Topics
Chapter 9. Trees
  • The matrix-tree theorem
  • Consequences of the matrix-tree theorem
  • Tree bijections
  • Rotor-routers
  • Notes
  • Problems
Chapter 10. Harmonic morphisms
  • Morphisms between graphs
  • Branched coverings of Riemann surfaces
  • Household-solutions to the dollar game
  • Notes
  • Problems
Chapter 11. Divisors on complete graphs
  • Parking functions
  • Computing ranks on complete graphs
  • Notes
  • Problems
Chapter 12. More about sandpiles
  • Changing the sink
  • Minimal number of generators for \(S(G)\)
  • \(M\)-matrices
  • Self-organized criticality
  • Notes
  • Problems
Chapter 13. Cycles and cuts
  • Cycles, cuts, and the sandpile group
  • Planar duality
  • Notes
  • Problems
Chapter 15. Matroids and the Tutte polynomial
  • Matroids
  • The Tutte polynomial
  • 2-isomorphisms
  • Merino's Theorem
  • The \(h\)-vector conjecture
  • Notes
  • Problems
Chapter 16. Higher dimensions
  • Simplicial homology
  • Higher-dimensional critical groups
  • Simplicial spanning trees
  • Notes
  • Problems
Appendices
Appendix A.
  • Undirected multigraphs
  • Directed multigraphs
Appendix B.
  • Modules
  • Chebyshev polynomials
List of symbols
Bibliography
Index